10 Billion Butterfly Sneezes

Guest post by Andi Cockroft(Anyone familiar with the Moody Blues should recognise the title – from “Higher and Higher”)

As some will have learned by now, I do not possess the scientific skills of the regular WUWT contributors to engage in in-depth evidential-based posts. Rather I like to think that just like the great unwashed masses, I have an intellect and an enquiring mind, and want here to share my musings and seek feedback to better help me (and hopefully other readers) understand some of the more complex subject matter. For my shortcomings I apologise. For raising questions requiring answers I do not!

Way back in January 2011, Phil Salmon posted here on WUWT “Is the ENSO a nonlinear oscillator of the Belousov-Zhabotinsky reaction type?” – His alternate title “Standing on the shoulders of Giant Bob” Perhaps my alternate title should be the same. I have read and re-read Phil’s definitive article, and as Isaac Newton was believed to have complained of the three-body problem:-“his head never ached but with his study on the moon”. So in my own way, I want to raise issues associated with what I see as true chaos, and whilst following (albeit very slowly) Phil’s work, that was nonetheless focused towards ENSO and away from the general Climate Models that I want to investigate.

Why a butterfly? Most students of chaos theory will know the “butterfly effect”, whereby it is asserted that in a chaotic non-linear system, a butterfly beating its wings in say Brazil could inject such feedback as to disrupt the airflow sufficiently that many years later a tornado would form over Texas.

Similarly, travelling back in time and moving the butterfly a few centimetres would cause the tornado not to form, or form elsewhere.

Also note that the Quantum version of the Butterfly Effect (the Quantum Butterfly), is a different beast altogether – although some parallels are drawn.

Ascribed to Edward Lorenz, the US mathematician and meteorologist who sadly passed away a few years ago. Initially Lorenz referred to a Seagull flapping its wings forever changing the weather, but later in a 1972 speech entitled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas”, the butterfly analogy was born.

It is Lorenz’s work on meteorological chaos that I want to look at specifically.

In or around 1961, vacuum-tube computers arrived at Lorenz’ disposal, and not surprisingly the first weather model was born as a set of a dozen or so differential equations involving such things as temperature, pressure, wind velocity etc.

During a re-run of this early model, Lorenz is believed to have restarted the program in the middle of its run by entering a variable to 3 decimal places – to his surprise the results were completely at odds with what was achieved earlier.

Restarting and re-entering the variable to its full 6 decimal places produced a repeat of the initial results – from this Lorenzo drew the inevitable conclusion that with his dozen or so equations, even a miniscule variation on input is capable of creating massive change in output.

But why such a radically different outcome for such a miniscule difference in input?

It transpires that the equations were non-linear, with a so-called “great dependence on initial conditions” – change the initial conditions even slightly and a large change in output is observed in a later state.

The conclusion that Lorenz drew, was that given that such small variations can create such massive variation in output, it was impossible to “model” a weather system.

This “sensitive dependence on initial conditions” was destined for higher things however.

Around this time, Lorenz published perhaps his most important paper “Deterministic Nonperiodic Flow”, and with it was born Chaos Theory, and the concept of the Lorenz Attractor.

(For those wanting to investigate the Lorenz Attractor further, I would recommend reading Phil’s original article here, or Wikipedia here )

I’m not sure if there is any relationship here to Peter, but James Gleick’s bestseller, ‘Chaos: Making a New Science’ built on Lorenz’ musings, became the mantra for many to follow in all walks of life:- finance, science and even time-travel effects in science-fiction. It is now bandied around in insurance, marketing and business boardrooms throughout the world.

Being quite a reserved individual, Lorenz was taken aback by the devotees to his speech. ‘I was just trying to determine why we didn’t have better luck with our weather forecasts, I never reached a point where I believed the butterfly was a scientific fact. At most, it’s a hypothesis. I never expected it to become so huge outside meteorology.”

So my question now is, given that we cannot even begin to measure things such as SST to anywhere near 3 decimal places, how can we expect low-accuracy and highly volatile data, entered into chaotic models of even more chaotic systems to produce anything of significance?

If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?

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75 thoughts on “10 Billion Butterfly Sneezes”

They really don’t care about most of their forecasts because they will be gone before they are proven right or wrong. It’s the MONEY now they care about and the blessings of their peers and politicians.

And what exactly makes policymakers so enamoured with the pronouncements of these modelmakers? It seems such a suicidal adherence. And the likes of Hansen and Mann, whose condescending machinations about nothing other than a total fantasy make such great cannon-fodder for even a street-level sceptic? (or great grist for the imagined legions of consensus science, some of whom populate the troll gallery at WUWT). All of this over a nonsensically-small delta-T, of which an even tinier amount “may” be caused by CO2. Totally wacky, I say.

It could be that climate and weather are something like Russian dolls from the dynamical systems viewpoint. That is, Lorenz says to expect weather to be a chaotic dynamical system on the time scale of days, which has a predictability horizon of a few weeks at the very best.

On a time scale of months, if you average the weather, your inner Russian doll begins to look like a single object because you can no longer follow the wanderings over the doll at that level. But on the next larger doll (say, ENSO) you may discern wanderings associated with another dynamical system. And it may be that, if you give up on weather forecasting, you could still find predictability of ENSO beyond the weather forecast horizon.

My understanding is that ENSO models show predictability beyond the weather horizon, but at the time scale of more than one ENSO cycle we are currently stumped.

But even that would not necessarily be the end of the story. It could be there is another doll (perhaps PDO) that can be understood as yet another dynamical system, after weather and ENSO are smoothed out by multimonth or multiyear averaging. What the forecast horizon of that might be is probably at the research frontier.

In order to claim climate forecasting skill, not only would these dolls need to be well understood, with a substantial portion of the decadal variability captured in the largest doll; you would also need to prove that GCMs can simulate them correctly. My understanding of Roger Pielke Sr’s and Judith Curry’s views is that we are very far from achieving that.

So skillful decadal forecasts are conceivable, but at this point it’s just as conceivable that, say, PDO dominates decadal variability, that PDO dominates the variability over the last 100 years; that PDO can be described as a dynamical system, but its major shifts cannot be forecast more that a few years before they happen. (Not saying that’s true, just that it can’t be ruled out with our present understanding.)

“If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?”
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The thing with these models and attractors is, just looking at the famous image (e.g. http://en.wikipedia.org/wiki/File:Lorenz.png) you can see that you cannot predict the exact point where will your system get after just a few circles, but you can predict pretty well the rough volume in which it will reside.
What models are used for is not to determine whether it will rain 100 years from now, but rather to determine the attractors around which the weather is oscillating and where will they move in the years to come.
Their skill in figuring that out is a completely different story but that does not have that much to do with chaos.

jhultquist, Ray Bradbury’s book is the first thing I think of when ever someone mention’s the “Butterfly Effect”
….…If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?….

As jack morrow said it is MONEY. But I would also add that it is POWER Dr. Evans explains it very nicely on JoNova’s Website.

Andi,I believe they have a valid answer to that.
Consider a different famous chaotic system, a swinging pendulum dangling from a swinging pendulum.
Although it’s chaotic nature means we cannot model where the pendulum will be beyond the very near future, we are still able to say a number of valid things about it. For example, we can determine the region it may be found within, and the region it can never be found within. We can expilicitly determine it’s location in the third dimension. We can determine the probabilities of it being found within particular bands of the regions that it may occur within.
In the same fashion, climate models can’t determine future weather. Not only does chaos mean they can’t determine snow won’t form in the Sahara desert, it means they can be sure that at some point it will. This is analgous to the bands that the pendulum may be found within. However, they can still say some extremely likely things about the temperature within the Sahara. They can say even more impressive things about the likely average temperature there over time. All this even though chaos does indeed mean they can’t tell for a particular day next decade whether that day will be a scorcher or a snowstorm.
So on this issue, I have to conclude ‘even a Climate Modeller can’t be wrong all the time’.
It’s not the chaotic nature of their models that troubles me. It’s not even my programmers distrust of the accuracy of very large programs. It’s the whole idea that the future can be foretold from the entrails of a computer.
All their models will be confounded if any of a number of possible things happens. Among the many possibilities are nuclear war between Israel and Iran, India and Pakistan, Russia and Georgia, China and India, or the U.S., or France, or England and whoever they deem themselves to be at risk from. Or a cometary impact. A supervolcano exploding. A new method of electricity production being developed. Natural or artificial changes in the biosphere that make it more carbon absorbant. Vastly improved battery storage would have the same effect as developing a new energy generation system. Changes in the economics of renewables. Changes caused by bioengineering or nanotechnology. Any of the unknowns that I haven’t mentioned. We don’t know what the future will be. But we do know the models do not foreshadow it.
Any idiot can forecast disaster with a computer and logarithmic growth, and many already have. Consider the forecasts of Ehrlich or the Club of Rome. Reality has always confounded them. And for the same reasons those earlier models were wrong despite all their claimed science and accuracy, these models too wiill be found wrong.
Of course, by then we’ll have moved on to the next prophesy of disaster. The late, respected G. Harry Stine, engineer and writer, anticipated it would be a fresh water shortage disaster.

I’d suggest enrolling in some advanced math courses at a good university. There’s far more to this than just sensitive dependence on initial conditions. And Lorenz’ work has been greatly expanded on. Univ. of Texas has a good curriculum. Blogs (even this one) cannot provide the depth and breadth of knowledge this subject requires.

As another meteorological novice, I understand your intuitive objectives to AGW “science” (really psudo-science).
Simple observations pointed out the fallacies inherent in making CO2 the sole determinate of future climate.
– It was obvious from my experience with western US evenings (vs. Southeastern US evenings) that water vapor, not CO2 is the real “greenhouse” gas. (Not the CO2 isn’t a greenhouse gas, it is just not very potent compared to the effects of water vapor and clouds.) Even my country bumpkin farmer relatives (with little or no education) knew that clear weather made for colder nights. However, we can model climate (to the 3rd decimal place) without really understanding cloud formation (as CERN experiments made clear) using the concentration of a trace atmospheric gas.
– Ignoring any possible effects of solar variability as being irrelevant to climate seemed ludicrous even though I no inkling of the real mechanisms by which the Sun affects climate.
– Assuming positive feedback mechanisms in any complex system that does not have a history of constantly going to extremes seemed stupid. Without negative feedback mechanisms in place it seemed obvious that any small variability would be amplified to the point of climate catastrophe every week, not just when man got involved.
I could go on but (hopefully) you get the idea.
CAGW is not just incorrect, it is based on so many obvious falsehoods that it should have been laughed at by people with any common sense long ago…except for the fact that it fits the prejudices (and political goals) of a large group of humanity (lets call them the “ruling elite” just for a handy nickname.) This ruling elite hates humanity (with the curious exception of their own group) and wants to punish (other) humans for destroying the perfect world that used to exist (sounds kind of like a religious story, except they HATE religion, go figure). Also, the ruling elite wants to control the lives of others and, oddly enough, CAGW gives them the perfect reason (either submit to our rule or you will be destroyed…Baw Ha Ha Ha!).
I salute those that work so hard to better understand the climate hoping that real science will discredit the pseudo-science of CAGW. However, I think that if facts would settle the argument, it would never have gotten going. Simply thinking about the climate, without really studying it should have made skepticism of CAGW the default position of all thinking people unless there were reasons of self interest for our beloved leaders to accept irrational arguments for taking our freedom (and our money). We will not win this argument with science because it was never about science (IMHO). That was obvious to this “casual” observer from the start.

“If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?”
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I don’t think they actually believe they can. What they believe is the cultist leftist economic and social dogma. They have to come up with something which fits their belief system….. thus climate modeling showing how rational humans, adapting and progressing through history, is evil.

This is an interesting and relevant post for “Climate Change”. Ten or more years ago I read an excellent book on Chaos Theory, which inevitably covered the story of Edward Lorenz and his work on the subject. He was truly a brilliant man who, but for WW2, would probably have become one of the greatest mathematicians of all time, instead of ending up as a meteorologist. (Sorry, Anthony!) I have always thought it unfortunate that the term “Chaos” became attached this theory. “Complexity” or “Complex” Theory seem to me to be more appropriate. In the same way, “Climate” is inappropriate when discussing the Earth’s changing ATMOSPHERIC conditions. In astronomy, for example, scientists never referred to the “climate” of a planet. “Climate” is only appropriate as a general description of regional annual weather conditions, viz “Mediterranean”, “Tropical”, “Temperate”, etc. And in each of these, considerable variations can occur throughout the year and by slight differences in geographical location. I reside in a small region of this planet called the British Isles, and we have great difficulty in keeping track of the variations in our “Temperate Climate” North to South, East to West, hour by hour. If the Earth has a climate, would someone kindly define it for me.

I’ve often thought it would be interesting to run one of these models with 128 bit floating point numbers and then repeat the exercise with 64 bit and 32 bit numbers and then compare the outputs. Any differences would highlight the futility in attempting to use models to predict a chaotic system.

I doubt that weather is completely chaotic in the long term. There are too many cyclic forces at work that are clearly observable. For example, we are in a 60 year ENSO cycle currently. The sun has an 11 year cycle. The duration of the natural cycles are not precise so may vary a bit.

My take is that mathematical equations of gases in the critical region may have zones of instability under certain process conditions. These zones of instability prevent the computer programs from converging. I recall one time when my department overhead shot up sharply one month. We were trying to resolve systems containing carbon dioxide. The Cray computers could not ever converge from one direction so would go the other way, but still could not converge. This back and forth was taking extremely high priced time on the Cray. I hired a retiree to come back and in short order he did the calculations near the zones of instability by hand and created convergence. Mathematics do not always reflect real world processes, especially when using computers to resolve. Computer models do not have as much judgment as knowledgeable humans.

The AGW models are so simplex that all driving forces except carbon dioxide do not have much impact. With carbon dioxide being a second order driver to water vapor, the models are essentially meaningless in the short run and doomed to complete failure in the long run. All they can do is to predict higher temperatures as the carbon dioxide content in the atmosphere increases. Actual observations of measured temperatures over the last 15 year prove that the models are incorrect as far as meaningful exactitude is concerned.

While the weather may jitterbug around chaotically, the temperatures are still limited by the conservation of energy. If the solar heat input doesn’t change, and the earth is really losing less heat to outer space (possibly because of GHGs), then the earth should get warmer. No telling for sure how the heat is distributed, but there is a tendency for things to mix.

Of course, this is an energy balance and an energy limitation; temperature is not conserved, energy is. If GHGs are adding energy to the earth, then it would not violate the conservation of energy for that to result in lifting the atmosphere further from the surface, without raising the tropospheric temperature. Indeed, energy could still be conserved if the additional energy went to increasing the speed of the earth’s rotation without raising temperatures at all. Theoretically.

If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?

I suspect that the Team have simply enlisted large numbers of volunteers to track the movements of all butterflies in Brazil and feed this data into the models.

Of course we could model things, but then the sun will decide to have two or three quiet cycles and suddenly everything is covered in ice, or a medium distant star will go nova and all the radiation balance assumptions of the model go right out the window. We live in a dangerous universe, but for sanity’s sake, we don’t dwell on the what-ifs of Alpha Centauri going nova on us and pretty much ending habitability in this region of space (probability ~nil, but the precautionary principle dictates we should take drastic action to see that such a catastrophic event does not occur /snark).

“Weather is not climate. To conflate the initial condition dependency problem of weather forecasting with the boundary condition problem of climate simulation is fundamentally erroneous.”

True, weather is not climate. However, climate modeling IS an initial value problem and the modeled climate WILL depend on initial conditions. This would be an excellent opportunity to discuss numerical stability and discretizations for system of differential equations, but no one from the warmist side is usually interested in those details…

Constrained infinities and constrained attractors make for good reading. The tip of a bull whip can crack at an infinite number of places but never farther away than the length of the whip. The reference frame can be moved from place to place such that it cracks in Bolivia one day and Tristan da Cunha the next. What actually happens inside the framework (weather) is not so important as what shapes the framework (climate) and what trends are possible. We have billions of years of weather history to ponder but we don’t have a clue as to what drives the climate to change and what it will soon become. We are trying to divine the characteristics and trend of the climate from the evidence given in the weather – learning the cause from observing the effect. We ignore Lorenz’s chaos and lessons of precision of initial starting points at our peril.

Kasuha says:
March 15, 2012 at 7:43 am
What models are used for is not to determine whether it will rain 100 years from now, but rather to determine the attractors around which the weather is oscillating and where will they move in the years to come.

That is not how the IPCC uses models. The current climate models are linear parametrized models super-imposed on weather forecasting (GCM) models. Thus, this article is correct in stating that current climate models cannot hope to predict the future climate.

Question to ponder: What were the initial conditions within the Singularity that lead to the Big Bang and the subsequent expansion of Spacetime? Because that’s where/when this needs to start. Good luck.

I went into atmospheric physics because of my interest in world development and the problem of hunger. It seemed to me that was the best thing I could do to help by contributing to or at least helping to deploy climate forecasts. I found out that was a vain hope, not least because of the stochastic complexity of the system. This was nearly 40 years ago. I spent some time in weather forecasting, and then an interest in air pollution led me to applied statistics, and by this time I had already decided that the major barriers to hunger reduction were political ones – the physical ones of energy production and agricultural and other techniques having been sufficiently reduced. And while I was engrossed in industrial statistics, the great climate crisis movement was getting underway big time. When at last I noticed it, only a few years ago, it was a bit of a Rip van Winkle moment for me. How could I not have noticed earlier!

It seems to hinge on CO2 having been declared by some to be a major driver of climate, a property which it has somehow failed to convincingly display in the past – a past which includes of course, far higher ambient concentrations of it. How has this happened? I suspect an unholy mix of unscientific thinking (more ‘geography’ than ‘physics’) and very effective political opportunism by such as the Club of Rome, mutually reinforcing in a very lucrative manner over the past 30 years or so. A key part of scientific method – the search for testable hypotheses and examining them by observation and experiment – seems to have been sidelined by emotive speculations, and graphic descriptions of climate doom, thanks to a ‘settled science’ giving such influence to a now much maligned molecule. The odious machinations of the IPCC have also played an important part in this sorry spectacle, with their specious urgency and alarming messages succeeding, it would seem, in suppressing critical thought in and around governments the world over.

Back in the late 1970s, it seemed to me that climate histories would have to suffice to give warning of possible conditions to come, and of course then there was superficial talk of a new glaciation just around the corner. My academic mentors did not take that at all seriously, and I do recall one lecturer being amused by the antics of a young Stephen Schneider who did. To my shame, I used to unsettle people in casual conversations by relaying the dramatic threat – not of ice sheets creeping down slowly from the north, but merely of the winter snow not going away over spring and summer. The ice was to grow up around us in due course, but in the meantime a covering of snow was more than enough to give us agricultural and other difficulties. I like to think I usually put them right about how speculative it all was, and that it was not in the least a real worry to me.

Now we have far more powerful computers, and we have had the breakthrough of a theory accounting for some of the covariance between sunspot cycles and weather records – the cosmic ray hypothesis by which these particles affect the rate of production of condensation nuclei relevant to cloud formation. A theory which seems to have been curiously disregarded by many, perhaps because it is not a suitable one for adding to the useful guilt of those pesky humans who are so detested by at least some of the people so keen on CAGW.

It seems to me still absurd to think of running grand climate models of the whole system by cycling through the simulated years and hoping for practical guidance by way of forecasts – the basic problem described by Lorenz remains. But perhaps there will be scope for elucidation of subsets of the system, say the regional impact of large mountains or extensive surface changes. Coupled with improved forecasts of magnetic field variations, more insights into the great ocean-related oscillations and other specific effects (such at those mountains), we might yet get our money’s worth out of those computers by their contributing more to the simple extrapolations of cycles and other trends. But I suspect the fixation on CO2 that has led to so much funding and so much fear will need to be dealt with first, coupled with a deliberate and sustained effort to make sure politicians in particular acquire a realistic expectation of what such models might do.

If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?

Because nature isn’t chaotic unless you insist on confusing it with mathematical models. In real life, “chaotic” merely means “really complicated”. Just because it’s hard doesn’t mean no one should try.

Not everything is caused by a butterfly flapping its wings. Some things are predictable from beginning to end – others have no known origin or termination point. If modelers were more honest about what they can’t predict we wouldn’t have reached this point.

“So my question now is, given that we cannot even begin to measure things such as SST to anywhere near 3 decimal places, how can we expect low-accuracy and highly volatile data, entered into chaotic models of even more chaotic systems to produce anything of significance?

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Small nit. We can indeed measure SST to three decimal places. The error is not in the measurement but in the presumption that such measurements no matter how aggregated can yield even so much a single digit representations of climate conditions.

The old saying goes that an economist knows the price of everything and the value of nothing.

Climate alarmists know the temperature of everything and nothing of the climate.

Surging and receding …
Surging and receding …
The sound of the waves rolling in and rolling back out has echoed across the world for hundreds of millions of years, a long reach toward eternity.
Not once in that span has it ceased rocking and crossing this blue world, sometimes gently, sometimes powerfully; stormy as the morning, calm as the deep of night.

Surging and receding …
Surging and receding …
The sea rolls in and rolls back out. One hundred billion shimmering stars rise between the wave crests, only to sink back into the vastness of the waves with the first dim light of dawn.
On a night of exceptional darkness, a faint shooting start cuts across the void, trailing a long tail of light, then falls behind the nacreous line of the horizon, its glow becoming an unfading scar-a memory in the space between the stars.
Gradually, the constellations change their shapes; white stars take the place of blue stars, red stars take the place of orange, each making way for the next as they slide past each other to weave new shapes in the sky.

Surging and receding …
Surging and receding …
Time that knows no haste flows over the waves as they roll in and roll back out, through the night and into day and into night again.

***

The vast flow of time leaves traces of its passage across everything without exception. It moves within everything that is, ……………………

From the Prologue of
Ten Billion Dats & One Hundred Billion Nights
by Ryu Mitsuse

Let’s say that you dump a box of 10,000 dominoes on your kitchen floor. Now balance one of them on its end. Then knock it down. Does this have an effect on a domino on the other side of the room? I don’t think so.

Now set up each of the dominos on their ends in some beautiful pattern that winds around the kitchen floor. Then knock one down. Does this affect a domino on the other side of the room? Much more likely, I think.

The climate of the planet is much more akin to the first domino experiment than to the second one. The climate, like the first domino experiment, is in an overall quasi-equilibrium. It is like a marble at the bottom of a bowl, not a marble balanced on the point of a needle.

Another point: Once the second domino experiment has been completed, the remaining pile of knocked-down dominoes is basically the starting condition for the first domino experiment. That is, a system may start out in an unstable situation (even an unstable equilibrium) but will tend to move to a stable one.

It is very difficult to get all the dominoes between the butterfly’s wings in Brazil and the air circulating in Texas lined up in such a way that the wing motion will trigger a tornado. This is because there are nearly an infinite number of dominoes between the two events.

On the other hand, a model can reduce the nearly infinite number of dominoes to just a handful of dominoes. Then there is a probability that the modeled dominoes will line up and a small effect at one end of the model will have a big effect at the other end.

In my opinion, it is this misunderstanding that causes many environmentalists to start with science and end up with Gaia. They believe the world is inherently unstable. Hence, any small change can have a profound effect – so all changes should be suspect and avoided. But how does the world exist in such an unstable state? Well, there must be a higher power (Gaia) that is keeping the dominoes properly aligned.

Andi: It is true that the weather/climate/atmospheric system of Earth exhibits nonlinear chaotic behaviors and that attempts to model-to-predict weather are severely hampered by sensitivity to initial conditions–a fact that comes more into play the longer the model runs, the further into the future (or past) one attempts to look. As I’m sure you know, weather forecasting is limited to about a week to ten days at best, after which it is a crap shoot.

But….like many nonlinear chaotic systems (NCS), there are known or assumed bounds and known or assumed ‘attractors’. For the Earth Climate System, northern hemisphere ice ages are considered by some to be one stable attractor with a fairly long temporal existence before the system shifts to a interglacial warm period–another considered attractor for the system.

As for bounds, it is generally accepted (but not proven in any scientific way) that Iceball Earth is not within the bounds of climatic variation and nor is Earth-as-Venus.

There are illustrations in James Gleick’s book of these concepts. For instance, on my laptop, I have Basic programs I wrote 20 years ago when I was toying with Chaos Theory. One of these draws a triangle. Depending on initial conditions, reiterating the formula through thousands of cycles draws a filled-in triangle (all points on the triangle and inside the triangle) OR points only on the outside lines of the triangle OR points only on the outside lines of the triangle and a particular ‘x’ shape inside the triangle. But ONLY these three choices….not any others. So this system has three possible states and definite bounds.

What do we know about the climate system and its possible attractor-states or its bounds? I’d say not that much — except for the direct observations of weather and climate in the historical period and the indirect–derived–observations about paleo-climate conditions.

I will add for the few commenters who read ‘nonlinear chaotic system’ (NCS) as ‘that weather is completely chaotic in the long term’.

The term Chaos for behaviors such as ‘extreme sensitivity to initial conditions’ is an unfortunate artifact of popular science. It does not mean that the resultant system itself is chaotic — in fact, nonlinear chaotic systems can be devilishly ordered but not predictable. Other systems show unpredictability/randomity within bounds over a wide range and then, in the middle of that, throw up a small patch that is finely ordered.

Because the movement of the planets is generally NOT a NCS, we can send a rover across a huge distance and time span to land on the surface of Mars. But because weather is, we can not place a bucket at a certain spot and catch a predicable bucket of rain water on a certain day one month from now.

Weather is not climate. To conflate the initial condition dependency problem of weather forecasting with the boundary condition problem of climate simulation is fundamentally erroneous.

Turn that around an think about it. If weather “is not climate,” yet we derive climate “data” from weather records, and assume that climate “determines” weather, then it follows that the initial conditions to which weather is sensitive are embedded in the climate system – if there is such a thing. That in turn means that the chaos that appears in “weather” must come from the climate system. If you instead consider “climate” as a long-term integral that derives from weather, then “climate” patterns and cycles are the product of oscillations typical of a system like Lorenz’s “butterfly.” That pseudo-cyclic pattern is scalable from daily, through annual to geological time scales from glacial epochs to much longer term patterns like the “hot house” – “ice house” pattern.

What may be fundamentally erroneous is assuming that climate is a “real” aspect of nature rather than an a reified (in the minds of climatologists) concept derived from an average experience of weather over time. Weather is real, but climate may be wishful thinking.

“During a re-run of this early model, Lorenz is believed to have restarted the program in the middle of its run by entering a variable to 3 decimal places – to his surprise the results were completely at odds with what was achieved earlier.

Restarting and re-entering the variable to its full 6 decimal places produced a repeat of the initial results – from this Lorenzo drew the inevitable conclusion that with his dozen or so equations, even a miniscule variation on input is capable of creating massive change in output.”

One would hope that he also tried it with the 3-digits and three trailing zeros, so as to see if it wasn’t an artifact of the number of digits input.

Three cycles of lunar declination tides from Christmas 2009 to march 8 2010 starting at 10 degrees North of the equator back to the same point.

The real chaos driver of the global circulation is the lunar declinational tides in the atmosphere, most of the energy of the declinational tidal effects is converted to meridional flow surges that manifest as Mobile Polar Highs, Rossby waves, or jet streams (call them what you want) but they are predictable from past patterns and make long range forecasting possible, with out their consideration numerical models hit the wall just past 7 – 10 days, as the tide turns every 13.6 days.

Start time and some of the jumps in the movie are due to problems with the GOES data base missing some of the picture needed, however the synchronizing is based on the same lunar declination angle on all three, per each frame with in a degree or so. The hope was to be able to show that there are repeating patterns in the global circulation because of the lunar declinational tidal effects acting on the atmosphere.

I live in New Zealand and forecast weather for cruising yachts. Weather as ‘Chaos’ causes us to smile as we have to endure the input of initial conditions when we evaluate the weather models we use daily (GFS,ECMWF, UKMet, Access, etc). A good forecaster will always look at the actual conditions and compare them to the model day 1 forecast. If they don’t agree we lose confidence in that model run. From a weather forecasting point of view Chaos to us is simply that the atmosphere is not linear and the main reason a model loses the plot is an error in the initial conditions. (not to mention that our understanding of weather and the modeling thereof is incomplete). Most of us use ensemble models now. Take the GFS, a robust and fairly good skill score model, tweak the initial conditions 30 or more times just ever so slightly and look at , for example, the overlay of SLP. If the isobaric lines all stack up nicely as the model forecasts forward in time we have confidence in the model run, if it looks like a plate of spagetti the confidence level falls off. In the SW Pacific we think models are good for 3 days, ok to 5 days and anything afterwards is an idea.

@Bruce Stewart 7:40 am:
“How is it conceivable that climate could be forecast years ahead?”

Oh, I am certain that it will be.

Look at so much that we have done with computers in each of the last 5-year periods since 1977. Not once in any of those would we have known what was coming in the next 5-year period. Certainly not in the 2nd one ahead.

How to do climate ahead? First we need to nail down the individual factors until they are repeatable. This will be done. But not by the dodos currently ‘running’ climatology.

There are a lot of those factors, so it will be some time before that is accomplished. But it will happen.

Secondly, someone will develop deterministic-Monte-Carlo math at a level we don’t know now. A math that handles truly complex problems. A new type of computer may even need to be developed to handle the complexity – it may not be just more HP or speed that is needed. That will be the day.

As a general rule, modernity sees itself as an apex. In reality it is a point along a continuum. Seen as an apex, there is nothing but void all around. When seen as a continuum, one sees that the present works with the past and the future. The ties to the future can be seen, but not from our present bottom-up scientific mindset, from which the apex-thinking derives. It can only be seen from the continuum mindset, if any.

That is what I see. It is amusing to see the floundering at present. But it won’t always be that way.

Rupert Sheldrake has a peek at all this, and so does Freeman Dyson have a glimmer. Sheldrake is a bit of a flake. No one has ever accuse Dyson of that!

But the math – that will come from someone like Mr Butterfly himself, Kerry Mullis, and his PCR — out of left field or riding in on a surfboard. I can’t wait to see it happen. Except it might be 125 years from now. But I don’t think so.

The short answer to the question is, they don’t.
Thats why they uses ensembles and take some mean vaue of all the results, even if they differ substantially.
In this way they get only the forcings, and all the computer models are just a smokescreen put on to hide this simple fact, which would not look so good when asking for more money and computers.

If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?

Because they have discovered an input parameter that completely dwarfs all other possible input parameters: CO2!

They are not very good at finding out if this input parameter is correctly used, but that would destroy all their certainty, so they ignore that.

There is another factor to make the climate models even more suspect. Climate models have positive feedbacks. That is, a relatively small change in inputs (e.g. greenhouse gases), result in a huge change in outputs (global surface temperatures and the stability of the weather) due to being amplified by other factors (e.g. water vapour). Mainstream economics models (based on similarly complex systems) assuming negative feedbacks (equilibrating systems) can still blow apart even with accurate data inputs. They need stringent “assumptions” about parameters (often a bit of common sense or standard economic assumptions) to keep in touch with the real world, especially the forecasting models.
Climate models have even more dogmatic assumptions, with a disdain for testing the accuracy of those assumptions, and an extreme haziness in reconciling the models with real world data. Look at the Skeptical Science website to see what I mean. Yet the reconciliation needs to be accurate. Why? Temperatures rose by around 0.7 Celsius in the 20th century, of which the anthropogenic factor was, at most, 0.4 C. The IPCC models reckon around 1.5C to 5.6C of anthropogenic warming this century, or at least 3.5 times and maybe more than 20 times the 20th century. So, even with nice linear, non-chaotic systems, the amplification factor is huge. Add in some chaos theory, and you end up with models scenarios little better than astrological charts.

Re : “Weather is not climate. To conflate the initial condition dependency problem of weather forecasting with the boundary condition problem of climate simulation is fundamentally erroneous.”

One could say that one butterfly’s wing flaps may ultimately produce an effect that increases the amount of energy radiated from the earth for a brief time and hence that butterfly was involved in a cooling event. Obviously different butterflys may be involved in warming events.

The effect of CO2 can be likened to a butterfly’s wing flaps too, but will it cause more cooling events, more warming events or be neutral to those events? AGW believers are strongly in the camp of neutrality and that the warming effect of CO2 is unaffected by the inevitable changes in day to day weather. That may be a valid view for the CO2 effect itself, but is it valid for its feedbacks too?

Interesting post! The thing about chaotic systems is that they are predictable in a statistical sense, but only as long as the parameters in the system are constant. Many chaotic systems are not only sensitively dependent on initial conditions, but also sensitively dependent on the parameters. The well known logistic map x(n+1)=r x(n)(1-x(n)) is chaotic for r=4 and has a very specific bath tub shaped statistical density (worked out by Ulam and Von Neumann) but as you reduce r the statistical distribution of trajectories changes completely. This is (one more reason) why climate modelling is so daft….you would need to know the governing equations and their parameters (which are also changing dynamically) as well as the initial conditions to infinite precision. The qualitative approach to climate prediction, looking at sun spots, ocean oscillations etc seems to me to be a far more fruitful approach than supercomputer number crunching approach. My two cents.

I believe the basic idea of the butterfly theory is incorrect.
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Not necessarily incorrect. But likely insufficient and simplistic. As is your domino theory. :)

Yes – necessarily incorrect in its literal form. The effect of a butterfly’s wings in Brazil will always be damped out long, long, long before it can affect the atmosphere in Texas, except in the case where the 10e30 “switches” between the wings in Brazil and the atmosphere in Texas have all been pre-aligned.

There is a reason that the examples of sets of differential equations that lead to Lorenz (strange) attractors that you may be familiar tend to be deceptively simple and have very few parameters. The more parameters of similar effect that a system has, the less likely that a discernible beautiful strange attractor will form. For a complex system with a huge number of parameters the Lorenz attractor may form if a small subset of those parameters tends to dominates the others. A “model” is an attempt to pare down the representation of a system to that small set of dominant parameters. The wings of the butterfly in Brazil do not quite qualify.

Models can be great for studying certain properties of complex systems. They may be good for identifying the existence various “strange attractor” type modes of the atmosphere. But they are lousy at predicting when the transition from one mode to another will occur, or what the trajectory through a particular mode may be at some arbitrary point along the evolution.

The general description for the butterfly effect is not what deterministic chaos is about. Deterministic chaos in non-linear dynamical systems is internally generated (intrinsic to the system), and not caused by random stochastic fluctuations caused by a butterfly. It is bound to the energy flow through the far from equilibrium system. Think of the formation of Benard cells, the temperature at which the cells appear is the bifurcation point.

Another great book about the subject is The Arrow of Time by Peter Coveney. Yes, that pesky Second Law and chaos seem to go hand in hand (irreversibility).

If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Simple – hubris and an agenda that relies on propping up AGW to sustain your ability to make a living as a “scientist”

Once again, we find that laymen’s use of the mathematics of chaos is a problem. The shame is that people think that if they understand an analogy of something then they understand that thing. Sorry folks but to understand chaos you have to understand the mathematics of chaos. Also, if one does not understand the mathematics, don’t tell people you do understand it.

Specifically the X-Y plot showing the initial conditions which look close to one another leading to four different final states, i.e. A, B, C, and D, does not show chaos. In fact, it shows the opposite of chaos, i.e. final states that have zero dimention. Chaos is not “sensitivity to initial conditions.” Chaos is that DESPITE a sensitivity to initial conditions, the trajectories tend trace out shapes that look very similar, i.e. Lorenz’s butterfly plot of his three variable system. No matter what initial condition is chosen (with the exception of the point 0,0,0, IIRC) the trajectories trace out shapes that look very much the same. The strange attractor is INSENSITIVE to the initial conditions.

“If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?”
Because the Climate Modelers believe that weather is statistical and Global Warming is not. It’s that simple. They analyze climate and think of weather as a perturbation. One could do all the analysis of weather systems and still not convince the GCM modelers that they are wasting their time.

“The conclusion that Lorenz drew, was that given that such small variations can create such massive variation in output, it was impossible to “model” a weather system.”
No. That is not what he said. We model weather systems all the time. There is a website called weather.com that receives millions of visits each day to get weather forecasts. They are quite successful. What Lorenz said was that LONG TERM modeling may be impossible. It is that kind of precision of thought which is not possible if one doesn’t understand the mathematics.

I am plagiarizing the information others have taught me, but here are some observations. Chaos is a form of stability. You are stable within the attractor, but cannot predict where within the attractor you will be. It is possible to leave the attractor and enter another with a different “shape” if the system conditions change sufficiently. For me, the primary question regarding deterministic chaos and climate is “What is the nature of climate sensitivity?” Paleo records are frequently used as evidence that sensitivity is significantly positive. So why do we believe sensitivity exiting the Younger Dryas was the same as it is now. Is sensitivity deterministically chaotic? I would be very surprised if it was either a constant, or linear and predictable. Therefore, I would be very surprised if we can ever model climate. We will never be able to do anything except make educated guesses based on our belief about the shape of the attractor.

“We can determine the probabilities of it being found within particular bands of the regions that it may occur within.”

No, that cannot be done. Probability theory and Chaos theory have nothing to do with each other because that doesn’t work. Back in the late eighties/early nineties when ‘Chaos: Making a New Science’ came out, I got very excited about that idea, went to one of my applied mathematics prof’s and asked him about it. He said no but I took a PhD level prob and stats class anyway just to figure it out for myself. What I got was a good understanding of theoretical prob and stats and no help what so ever in modeling chaotic systems.

At a more fundamental level “the regions that it may occur in” can’t be defined and hence one can’t properly box the strange attractor. It’s the classic Mandelbrot “How long is the coast of Britain” problem.

Too many things left off the table – for example:
1. fundamental difference between temporal chaos & spatiotemporal chaos.
2. strange nonchaotic attractors.
3. global constraints indicated by EOP …begs questions about aggregation spatiotemporal scales – very serious stuff that (very unfortunately) flies over, around, or through (in one ear out the other) the heads of ~99.999% of the people involved in the climate discussion.

Going severely wrong is as simple as making an untenable base assumption. FOR SURE there are mainstream misconceptions which have run wildly unchecked in an abstract vacuum. Spatiotemporal chaos in a shape-shifting box – where solar & lunisolar cycles DO constrain the shape-shifting in a previously underappreciated manner – might be the closest thing to common ground that can be used as a practical conversation starting point. (It’s foolish to waste so much as a single breath trying to frame terrestrial climate as temporal chaos.)

I think the reason these models “work” (ie don’t show chaotic outputs) is that they do something like this:

Let i=1,…n denote a cell (ie position on the earth’s surface) and let T(i,t) be the temperature in cell i at time t. The equation for CO2 (concentration C, same for all cells) “forcing” is something like
T(i,t+1) = T(i,t) + kf(C)
for some parameter k and function f. If heat capacity is the same for all cells the mean global temperature G(t) is given by a simple mean over all i=1,n of the T’s and
G(t+1) = G(t) + kf(C)
Now apply a “mixing equation” which stirs things up by shufffling the i’s about (eq T(i,t) becomes the new T(j,t) etc).; more generally each T(i,t) is some linear combination of the others). If we now take the mean of the new temperatures we are getting the mean of the same numbers as before, we have just changedd the order). So the mean global temperature G is the same as before mixing.

That is, mixing does nothing to obscure the important part, kf(C). So if you incorporate some sensible constraints (eg conservation off energy) to prevent chaotic collapse it does matter what mixing equation is used (remember, they make no claim to local prediction, only global)

A fractal model is what is required – but the computing power required would be enormous. Might as well get some mice to build another planet just for testing. Just make sure we get to the Vogon planning office in time.

@davidc – Google “Hadley Cell”. On a side note, the Hadley cell on an integral scale is intellectually similar to Lorenz’s cell on a differential scale. Lorenz simplified the “mixing equation” to a simple three variable system. He then tried to numerically integrate it and found that he could not get stable solutions, i.e. no points A, B, C and D.

Weather is not climate. To conflate the initial condition dependency problem of weather forecasting with the boundary condition problem of climate simulation is fundamentally erroneous.
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If the climate models are solving boundary value problems, i.e., the solution is a function of the spatial variables, then it’s mathematically impossible for the climate models to predict the future.

The Navier-Stokes equation is a boundary value problem with initial conditions.

In fact, you can view the Lorentz equation (the 3 mode truncation of the Benard heat convention problem) as a metaphor for the Navier-Stokes equation in 2 spatial dimensions.

More formally, the effect of mixing on global mean temperature:
T(i) = temperature in cell i befoer mixing; n cells
H(i)=CpT(i) = heat content of cell i before mixing; Cp=heat capacity, same for all i
G = global mean temperature before mixing = (1/n)[sum over i of]H(i)/Cp
After mixing the heat content of cell i, Hmix(i) is
Hmix(i) = [sum over j of]L(j,i)H(j)
where L(j,i) is the fraction of the heat content of cell j transfered to i (including itself; ie heat retained by cell i).
Conservation of heat during mixing:
[sum over i of]L(j,i) = 1;
that is the total of heat transfered from cell j to all other cells (including itself; ie heat retained) is the original heat content of cell j.
The global mean temperature after mixing, Gmix, is
Gmix = (1/n)[sum over i of]Hmix(i)/Cp
=(1/n) [sum over i of][sum over j of]L(j,i)H(j)/Cp
= (1/n)[sum over j of][sum over i of]L(j,i)H(j)/Cp; reverse of of the two summations
= (1/n)[sum over j of] H(j) [sum over i of]L(j,i)/Cp; H(j) independent of i
= (1/n)[sum over j of]H(j)/Cpas G before mixing
since by conservation of heat [sum over i of]L(j,i)=1.
But this expression for Gmix is the same as the expression for G (before mixing; except for the index j in place of i).

I can’t think of a way to describe mixing such that the outcome is not a linear combination of the original values, but maybe someone can point out my error. If I’m right then whatever equation they use for mixing will have no impact on the global mean temperature calculated. So it is determined solely by the assumed forcing equation for the effect of CO2 on temperature.

If the models are written with internal “constraints” to stop them leaving the rails, then how much of this mathematics becomes a moot point? Protein-folding models are far simpler systems yet exhibit the same problems.

Dinostratus, thanks for the reference. From Wikipedia this 1963 one is on atmospheric circulation (I haven’t found the original yet). My point is that if the sole object of the modelling is to predict the mean global average temperature (which I believe the IPCC says; no modelling of local effects) the method of modelling the atmospheric circulation (“mixing”) does not matter. Whether this is by Lorenz, Navier-Stokes or anything else it all comes out the same. The key equation is of the form
G(t+1) = G(t) +kf(C)
where the advance of the global average fro time t to t+1 depends only on the assumed temperature change in response to CO2. I’m assuming a computational scheme:

My point is that Step 2 (eg Lorenz, Navier-Stokes) makes no contribution to the final result for is smoke and mirrors: this is too hard for you to follow, just trust me.G.
If you think I’m trying to establish an awesome generality for the models, I’m trying to establish the opposite: that they are trivial. For example over the period from about 1970 to about 1998 the reported global average temperature increased approximately linearly as did CO2 levels. So in place of the general term kf(C) we have a linear term to give something like
G(t+1) = G(t) + a + b.deltaC
where deltaC is the change in C over the time interval (t,t+1). So, simple linear regression is what’s needed to find a and b. I think the only function of the GCM is smoke and mirrors (this is too hard for you, just trust me).

If the models are written with internal “constraints” to stop them leaving the rails, then how much of this mathematics becomes a moot point? Protein-folding models are far simpler systems yet exhibit the same problems.

;———————————————————————————————–

The motion is determined by the driving and diffusive forces.

It’s also a plot in phase space.

The major deficiency of the Lorentz model is the extreme truncation to 3 modes.

There are 14 modes so the attractor has a set of rich dynamics.

It turns out the most sensitive parameter of the Lorentz equation is the number of modes.

“this is too hard for you to follow, just trust me.G.”
No problem. I first saw the NS equations in their full glory over twenty years ago. That’s not enough time to understand them so I’m still working on it.

“My point is that if the sole object of the modelling is to predict the mean global average temperature (which I believe the IPCC says; no modelling of local effects) the method of modelling the atmospheric circulation (“mixing”) does not matter.”
That may be one small step for you but it is a giant leap for mankind. Your assumption is that the “mixing” doesn’t matter. Your equations then show that it all comes down to other variables then you conclude that the mixing doesn’t matter. In fact, the integration of the mixing terms is nontrivial.

What Al Gore’s professor, Dr. Revelle, showed was that for a zero dimensional model such as yours, global warming happens as their is an amplification cycle of human generated CO2 to water vapor, Since then everyone’s been arguing over the water vapor issue. More detail has been added to the models and hence fewer assumptions need to be made, i.e. mixing models, etc.

“the method of modelling the atmospheric circulation (“mixing”) does not matter”
That is the fundamental question that Lorenz was asking. He modeled a differential buoyancy driven element to determine if the mixing could be integrated. He thought not (obviously) but others have come along making assumptions that overcome Lorenz’s difficulties. Whether those assumptions are valid or not is up for debate. Sometimes we come up with very good mixing models and sometimes we don’t. Over the years I’ve noticed that when simplifications of the NS equations retain a coupling between the elliptical terms and parabolic terms, our ability to model the flow is poor.

An analogiy might help to make my point. I am drinking a cup of cofee and I have added a teaspoon of sugar and stirred it with the teaspoon. I could in principle apply the Navier-Stokes equation to describe what happens as I stir (tricky, time dependent boundary conditions), or some other model. As a scientist I insist that my model (of stirring/mixing) be tested against observations. But I also insist that the observation that is to be used to test my model is the average concentraion of sucrose (sugar) over the entire cup. Using concentration units of teaspoons/cup we know what the answer will be: 1 teaspoon/cup. To get this answer the stirring/mixing model has to do just one thing: satisfy conservation of mass. If it does this by any means at all (Navier-Stokes, something to do with Lonenz, patterns in tealeaves constrained somehow to satisfy conservation of mass) then when I add up all the little bits of mass being stirred around I get 1 teaspoon. The volume of the cup remains at 1 cup. The average concentration (total mass/volume) is then 1 teaspoon/cup. So the model is validated (all of them, and any stirring model that satisfies conservation of mass that you or anyone else might come up with).

I think I understand. The sugar in your example is called a passive scalar. “Passive” because it does not affect the mixing mechanics. What you describe is called a “Stirred Reactor”. The simplest reactor model is a “Perfectly Stirred Reactor.” The test to see if a reactor is perfectly stirred is the Damkohler Number. If the reaction rate is slow compared to the mixing rate we say the Dh<<1 and the reactor is perfectly stirred and the fluid mechanics don't need to be modeled. We just assume that the reactor is homogenous.

The giant leap you are making is the assumption that the sugar is a passive scalar.

Temperature, in the case of the atmosphere, is NOT a passive scalar. It is the temperature gradient that drives the buoyancy driven flows (not directly but through density). This is important because it is the temperature at the surface that drives the liberation of CO2 and water vapor from the oceans. The faster the convection (or "advection" in the language of atmospheric scientists), the cooler the surface and the less liberation of CO2. Moreover, the higher the surface temperature, the more water vapor is loaded into the atmosphere and then carried up to make clouds.

Spencer has spent a lot of time on the last issue and it is that water vapor / cloud model that is (and always has been) the weakest link of GCM models.

Using your analogy,if the sugar is an active scalar then the sugar is also a spoon and makes sugar (or not) as it settles to the bottom. One can't say for sure that there is only one teaspoon of sugar.

PS – If you want to read something really whack, read the comments on global warming by Octave Levenspiel, the author of "The Chemical Reactor Omnibook" which is the seminal reference on reactor modeling.

Dinostratus, Yes, the analogous quantity should be heat, but since the heat capacity will be constant this doesn’t alter the argument in a significant way.

But you are misssing my main point which is the use of the climate models to estimate global mean temperature, not local temperatures. We are told that the models don’t predict local effects. Some of us might think it’s odd that you could calculate a global mean temperature for the model without knowing local values (after all that is how they estimate the so-called “observed” global mean temperature). My suggestion here is that that is how it’s estimated in the models but the local effects in the models don’t correspond to any real local effects. But – main point – if you stick to global mean temperature as the value reported it is irrelevant whether the model works at that level or not (provided it satisfies some conservation of heat constraint).

Back to the coffee. Suppose you added the sugar as a syrup containing the equivalent of one teaspoon of solid sugar, carefully layered on the bottom. The appropriate model for “mixing” would then be the diffusion equation. The distribution of sugar in the coffee is quite different from the stirred example. But if you define the average concentration in the cup in the obvious way as amount/volume, solved the diffusion equation for some value of time, integrated the concentration-position profile to get an average concentration, you would get … 1 teaspoon/cup. If you did the calculation with the wrong diffusion coefficient it wouldn’t matter, still 1 teaspoon/cup. If you made a mistake and used 1/D where it should be D, again doesn’t matter, still 1 teaspoon/cup.

The only way to see which model is correct is by sampling local concentrations (say, by drinking the coffee). The stirred model would clearly be wrong when you find that the first few mouthfuls were not sweet enough, the last few much too sweet (but on average, just as you like it).

If temperature is sugar then the assumption you are making is that the sugar/temperature is a conserved passive scalar. In the case of the atmosphere, temperature is not conserved. It is increased or decreased depending on the temperature gradient. It is not passive. Buoyancy drives the mixing rate. Energy is conserved. Not temperature.

Just because temperature can be integrated doesn’t mean it is conserved. One can integrate the velocity in the coffee cup too. It integrates to zero (assuming you let it sit on a table and haven’t thrown it at me). Velocity however is not conserved.

You really need to read Lorenz’s paper. You’re asking good questions. Essentially he tries to numerically integrate a differential cell. He finds that he can’t because of the deterministic non-period flow that the model suggests.

Again, using the syrup example, the syrup (temperature) gradient creates (or destroys) syrup (temperature). The strength of the temperature gradient either liberates or traps heat, depending. It’s not the instantaneous integration that is the problem. It’s the time varying evolution of those integrations that causes the problem. Taking your equation, G(t+1) = G(t) +kf(C), the problem is that k depends on f(CO2, H2O, T, dT, etc.). k depends on clouds, H2O(l) & H2O(s), and clouds are not modeled well at all.

As far as the CGM models, no they don’t get local results right. (Curry has gained an undeserved reputation as an skeptic because she cautions not to predict local weather changes using CGM’s.) However the modelers ASSUME that the differences are statistical in nature. That is, they average out to zero (the Central Limit Theorem applies). Do they? Lorenz says no. Tennekes has a good paper on that issue.

I do agree with one of your main points. The CGM folks use models that are insensitive to individual cell variations. Cell variations are damped out to the point that the computers can make projections. New computers are then purchased, the models are lessened up a bit and then re-run on faster computers. It’s a nice budget scheme for National Lab and University managers. They did the best they could with what they had and would like more capability to do better. Cha-ching.